For any \(\lambda > 0\) , denote \(\mu _2^b(\lambda ) = \mathrm {var}(\hat{r}^b_{\lambda })\)  as the variance of the estimated rank from the reduced rank regression based on the subsample \((Y_1,X_1),\ldots ,(Y_b,X_b)\) . The parameter \(\mu _2^b(\lambda )\)  can be written as \(\mu _2^b(\lambda ) = m_2^b(\lambda ) - \lbrace m_1^b(\lambda )\rbrace ^2\) , where \(m_2^b(\lambda ) = \mathrm {E}[(\hat{r}^b_{\lambda })^2]\)  and \(m_1^b(\lambda ) = \mathrm {E}(\hat{r}^b_{\lambda })\) . Note that \(\hat{m}_1^b(\lambda ) = \sum _{i=1}^N \hat{r}_{\lambda }(S_i)/N\)  and \(\hat{m}_2^b(\lambda ) = \sum _{i=1}^N \hat{r}^2_{\lambda }(S_i)/N\)  are their corresponding U-statistics of order \(b\) , and further that \(0 \le \hat{r}_{\lambda }(S_i) \le r_{\mathrm {x}} \wedge q\)  for \(i=1,\ldots ,N\) . Hence, by Hoeffding's inequality for U-statistics [1]}, we have, for any \(t > 0\) ,
\(P\left[\left| \hat{m}_1^b(\lambda ) - m_1^b(\lambda ) \right| > t\right] & \le 2 \exp [-2 n t^2/\lbrace (r_{\mathrm {x}} \wedge q)^2 b\rbrace ], \\P\left[\left| \hat{m}_2^b(\lambda ) - m_2^b(\lambda ) \right| > t\right] & \le 2 \exp [-2 n t^2/\lbrace (r_{\mathrm {x}} \wedge q)^4 b\rbrace ]. \) 
